| L(s) = 1 | − 5·2-s + 5·3-s + 3·4-s + 6·5-s − 25·6-s + 27·8-s − 49·9-s − 30·10-s + 2·11-s + 15·12-s + 80·13-s + 30·15-s − 37·16-s + 39·17-s + 245·18-s + 24·19-s + 18·20-s − 10·22-s + 120·23-s + 135·24-s − 255·25-s − 400·26-s − 353·27-s − 184·29-s − 150·30-s + 4·31-s − 53·32-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.962·3-s + 3/8·4-s + 0.536·5-s − 1.70·6-s + 1.19·8-s − 1.81·9-s − 0.948·10-s + 0.0548·11-s + 0.360·12-s + 1.70·13-s + 0.516·15-s − 0.578·16-s + 0.556·17-s + 3.20·18-s + 0.289·19-s + 0.201·20-s − 0.0969·22-s + 1.08·23-s + 1.14·24-s − 2.03·25-s − 3.01·26-s − 2.51·27-s − 1.17·29-s − 0.912·30-s + 0.0231·31-s − 0.292·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + 5 T + 11 p T^{2} + 17 p^{2} T^{3} + 11 p^{4} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 - 5 T + 74 T^{2} - 262 T^{3} + 74 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 6 T + 291 T^{2} - 988 T^{3} + 291 p^{3} T^{4} - 6 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 2713 T^{2} + 13108 T^{3} + 2713 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 80 T + 7775 T^{2} - 355808 T^{3} + 7775 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 39 T + 10764 T^{2} - 269068 T^{3} + 10764 p^{3} T^{4} - 39 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 24 T + 19881 T^{2} - 312456 T^{3} + 19881 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 120 T + 28569 T^{2} - 1922592 T^{3} + 28569 p^{3} T^{4} - 120 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 184 T + 39951 T^{2} + 3957616 T^{3} + 39951 p^{3} T^{4} + 184 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 6029 T^{2} - 283720 p T^{3} + 6029 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 589 T + 7220 p T^{2} + 67144276 T^{3} + 7220 p^{4} T^{4} + 589 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 92 T + 4027 p T^{2} - 7995456 T^{3} + 4027 p^{4} T^{4} - 92 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 250 T + 171365 T^{2} + 42748948 T^{3} + 171365 p^{3} T^{4} + 250 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 459 T + 280872 T^{2} - 63986692 T^{3} + 280872 p^{3} T^{4} - 459 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 579 T + 228972 T^{2} + 94125566 T^{3} + 228972 p^{3} T^{4} + 579 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 267 T + 543684 T^{2} + 130418348 T^{3} + 543684 p^{3} T^{4} + 267 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 540 T + 119601 T^{2} - 142838224 T^{3} + 119601 p^{3} T^{4} + 540 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 749 T + 628750 T^{2} - 200987390 T^{3} + 628750 p^{3} T^{4} - 749 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 1924 T + 2114435 T^{2} - 1560841600 T^{3} + 2114435 p^{3} T^{4} - 1924 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 805 T + 1187566 T^{2} - 671146982 T^{3} + 1187566 p^{3} T^{4} - 805 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 712 T + 1422097 T^{2} + 602850736 T^{3} + 1422097 p^{3} T^{4} + 712 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 835 T + 2039856 T^{2} + 1065240352 T^{3} + 2039856 p^{3} T^{4} + 835 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2243 T + 3813620 T^{2} - 4231696160 T^{3} + 3813620 p^{3} T^{4} - 2243 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.324919464251348796896901887415, −7.76601964139903651133305955627, −7.74980994373844240235809571051, −7.65218899324940553793650283656, −7.18987530338448402264622565424, −6.62172767793934789756910731249, −6.60813837554559710845906920739, −6.17151983094887486846541122730, −5.93310996012486128070992213478, −5.73165856460504674625132407322, −5.40444136842827528059477395603, −5.12118387918559621410389907174, −5.04822213992841501456209651481, −4.54711451831196728018899858795, −3.83207613261055745344395426308, −3.77128422246405960976960388949, −3.60439659679719280749061617265, −3.24232469971577685779499603281, −3.10801543592571737901482274284, −2.47054164739001322368043996366, −2.23603368440831397540480258419, −1.89305390626045416237344965601, −1.53185397192955000362567362932, −0.976858488604196520562315548070, −0.961879210842289549087562726440, 0, 0, 0,
0.961879210842289549087562726440, 0.976858488604196520562315548070, 1.53185397192955000362567362932, 1.89305390626045416237344965601, 2.23603368440831397540480258419, 2.47054164739001322368043996366, 3.10801543592571737901482274284, 3.24232469971577685779499603281, 3.60439659679719280749061617265, 3.77128422246405960976960388949, 3.83207613261055745344395426308, 4.54711451831196728018899858795, 5.04822213992841501456209651481, 5.12118387918559621410389907174, 5.40444136842827528059477395603, 5.73165856460504674625132407322, 5.93310996012486128070992213478, 6.17151983094887486846541122730, 6.60813837554559710845906920739, 6.62172767793934789756910731249, 7.18987530338448402264622565424, 7.65218899324940553793650283656, 7.74980994373844240235809571051, 7.76601964139903651133305955627, 8.324919464251348796896901887415
